Single Particle Excitations in the Lattice E₈ Ising Model

We present analytic expressions for the single particle excitation energies of the 8 quasi-particles in the lattice $E_8$ Ising model and demonstrate that all excitations have an extended Brillouin zone which, depending on the excitation, ranges from 0<P < 4\pi to 0< P< 12 \pi. These are compared with exact diagonalizations for systems through size 10 and with the E_8 fermionic representations of the characters of the critical system in order to study the counting statistics.